The geometric sequence $(a_i)$ is defined by the formula: $a_1 = -\dfrac{8}{3}$ $a_i = \dfrac{3}{4}a_{i-1}$ What is $a_{4}$, the fourth term in the sequence?
Solution: From the given formula, we can see that the first term of the sequence is $-\dfrac{8}{3}$ and the common ratio is $\dfrac{3}{4}$ To find the fourth term, we can rewrite the given recurrence as an explicit formula. The general form for a geometric sequence is $a_i = a_1 r^{i - 1}$ . In this case, we have $a_i = -\dfrac{8}{3} \left(\dfrac{3}{4}\right)^{i - 1}$ To find $a_{4}$ , we can simply substitute $i = 4$ into the formula. Therefore, the fourth term is equal to $a_{4} = -\dfrac{8}{3} \left(\dfrac{3}{4}\right)^{4 - 1} = -\dfrac{9}{8}$.